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Second-Order Hydrodynamics and Universality in Non-conformal Holographic Fluids

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Applications of the Gauge/Gravity Duality

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Abstract

This chapter contains our work on second-order non-conformal hydrodynamics. We first derive five new Kubo formulae that express second-order transport coefficients in terms of three-point correlators of the stress tensor. We then apply these Kubo formulae to a large class of non-conformal holographic fluids at infinite coupling. We find strong evidence that the Haack–Yarom identity [14], which relates second-order coefficients in conformal holographic fluids at infinite coupling, continues to hold for holographic fluids without conformal symmetry. Within our class of models, we prove that the identity is still obeyed when taking into account leading non-conformal corrections, and show numerically that the identity continues to hold further away from conformal symmetry.

This chapter is based on P. Kleinert and J. Probst, Second-Order Hydrodynamics and Universality in Non-Conformal Holographic Fluids, JHEP 12 (2016) 091, [1610.01081] (Ref. [1]).

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Notes

  1. 1.

    Essentially because all (non-topological) matter couples to the volume element \(\sqrt{-g}\).

  2. 2.

    Other recent holographic and non-holographic studies of second-order transport in non-conformal relativistic fluids include Refs. [52,53,54,55,56,57,58].

  3. 3.

    In both cases the bulk geometry can be viewed as a compactification of AdS and one can essentially borrow the higher dimensional AdS/CFT dictionary [59, 60].

  4. 4.

    Note that if we wanted to extract all fifteen second-order coefficients we would have to turn on metric perturbations in the scalar sound channel, too. Such perturbations, however, would necessarily source fluctuations of \(\delta \epsilon \) and \(\underline{v}\).

  5. 5.

    Eq. (4.21) constitutes the extension of the linear-response result (3.17) to quadratic-response. The factor of \(\frac{1}{2}\) in the first line of Eq. (4.21) avoids double-counting of the symmetric \(h_{\rho \sigma }=h_{\sigma \rho }\). The factor of \(\frac{1}{8}=\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2}\) in the second line avoids double-counting of the symmetric \(h_{\rho \sigma }=h_{\sigma \rho }\) and \(h_{\kappa \lambda }=h_{\lambda \kappa }\), and accounts for the term representing the second-order expansion.

  6. 6.

    Ref. [7] defines the Fourier-transformed three-point functions with the opposite sign for the two momenta. In our convention, the shear viscosity is given by \(\eta =i\left.\partial _{q_0}G^{xy,xz,yz}(q,p)\right|_{q=p=0}\), as opposed to Eq. (21) in Ref. [7], \(\eta =-i\left.\partial _{q_0}G^{xy,xz,yz}(q,p)\right|_{q=p=0}\). We further believe that the factors of 2 in their Eqs. (22) and (23) should be absent, in agreement with their Eq. (26).

  7. 7.

    Given that the bulk metric will be diagonal to leading order, \(g_{mn}\propto \delta _{mn}\), we can ensure that \(\mathrm {EOM}^{xy}=\mathcal {O}(\epsilon ^3)\) by solving the usual form of Einstein’s equations with lower indices, \(\mathrm {EOM}_{mn}\), to \(\mathcal {O}(\epsilon )\) and \(\mathrm {EOM}_{xy}\) to \(\mathcal {O}(\epsilon ^2)\).

  8. 8.

    We restrict ourselves to operators of dimension \(\Delta =3\) because the holographic renormalisation has already been done for this class of holographic RG flows. See Appendix C of Ref. [1] for details.

  9. 9.

    Inserting (4.38) into (4.32) and switching to the radial coordinate \(z=\sqrt{u}/\left(\pi T\right)\), the \(AdS_5\) black-brane geometry takes the more familiar form \(\mathrm {d}s^2=\frac{L^2}{z^2}\left(\frac{\mathrm {d}z^2}{f} -f \mathrm {d}t^2 +\mathrm {d}\underline{x}^2\right)\) (see e.g. Eq. (2.15)).

  10. 10.

    It is for this technical reason that we restrict ourselves to perturbations in the transverse shear channel and do not consider metric perturbations in the scalar sound channel. The drawback of this restriction is that we only gain access to five independent combinations of transport coefficients, Eq. (4.2), as explained in Sect. 4.2.1.

  11. 11.

    Moreover, \(H^{(1t)}\) and \(H^{(1z)}\) are again normalised to 1 at the boundary and, as we will discuss in Sect. 4.4.2, are subject to the same boundary conditions at the horizon as they are in cases (4.42) and (4.46). Hence \(H^{(1t)}\) and \(H^{(1z)}\) here are indeed the same as in perturbations (4.42) and (4.46).

  12. 12.

    Provided the holographic RG flow is indeed described by a black-brane geometry with a horizon, Eq. (4.32) (rather than by a confining geometry, for example).

  13. 13.

    Curiously, it turns out that the one hydro metric fluctuation for which we did not find a solution cancels out in the expression for \(\left<T^{\mu \nu }\right>\) presented in Sect. 4.5.1.

  14. 14.

    Note that potentials \(L^2\,V=-12-\left(3/2\right)\phi ^2+\mathcal {O}\left(\phi ^4\right)\) which follow from an appropriate superpotential \(L\,W=-\left(3/2\right)-\phi ^2/8+\mathcal {O}\left(\phi ^4\right)\) via \(V=8\left((\partial W/\partial \phi )^2-(2/3)W^2\right)\) automatically satisfy condition (4.57). We believe that condition (4.57) was overlooked in Ref. [50].

  15. 15.

    There are two first-order fluctuations \(a\in \big \{1t,1z\big \}\), each with three relevant hydro series coefficients \(j\in \big \{0,1,2\big \}\), as well as three second-order fluctuations \(a\in \big \{2tt,2zz,2tz\big \}\), each with six relevant hydro series coefficients \(j\in \big \{(0,0),(1,0),(0,1),(2,0),(1,1),(0,2)\big \}\).

  16. 16.

    Provided the RG flow is holographically described by a black-brane geometry with a horizon, Eq. (4.32).

  17. 17.

    In the conformal case \(\phi =0\), Eq. (4.38), this reduces to \(\bar{\epsilon }=3\bar{p}=\frac{3\pi ^3L^3}{16 G_N}T^4\) or, for \(\mathcal {N}=4\) specifically where \(l_P^8=G_N\,L^5\text {Vol}(S^5) =\left(\pi ^4L^8\right)/\left(2N^2\right)\), \(\bar{\epsilon }=3\bar{p}=\frac{3\pi ^2}{8}N^2T^4\)  [80].

  18. 18.

    We chose to explicitly state the result for \(\lambda _1+\lambda _3/4\) (rather than for \(\lambda _1+\kappa ^*/2\) or \(\lambda _3-2\kappa ^*\)) because \(\lambda _3\), contrary to \(\kappa ^*\), does not vanish for conformal fluids in general [7, 64] and thus allows for a more meaningful comparison with the conformal case. It does, however, vanish in conformal holographic theories at strictly infinite coupling [24].

  19. 19.

    The fact that the number of degrees of freedom at an \(AdS_{d+1}\) fixed point with AdS radius \(\ell \) scales with \(\ell ^{d-1}\) can be seen from the Weyl anomaly [81]. The holographic Weyl anomaly \(\left<T^\mu {}_\mu \right>\) is given by a surface contribution of dimension d, multiplied by \(1/G_N\) times the appropriate power \(\ell ^{d-1}\) of \(\ell \) to ensure that the Weyl anomaly has the correct dimension d. Because \(\left<T^\mu {}_\mu \right>\) is directly proportional to the central charge, the number of degrees of freedom, too, scales with the AdS radius as \(\ell ^{d-1}\).

  20. 20.

    The fact that \(c_s^2=\mathrm {d}\bar{p} / \mathrm {d}\bar{\epsilon }\) can be written as \(c_s^2=\left(\mathrm {d}\log T\right)/\left(\mathrm {d}\log s\right)\) follows from standard thermodynamic relations. The first law for an uncharged system with energy E and entropy S reads \(\mathrm {d}E=T\,\mathrm {d}S - \bar{p}\,\mathrm {d}V\). Using the extensivity of E, S, and V, combined with the homogeneous function theorem, gives \(E=T\,S-\bar{p}\,V\). This yields \(0=S\,\mathrm {d}T - V\mathrm {d}\bar{p}\), implying \(\mathrm {d}\bar{p}=s\,\mathrm {d}T\), as well as \(\bar{\epsilon }+\bar{p}=Ts\). Combining the last two relations gives \(\mathrm {d}\bar{\epsilon }=T\,\mathrm {d}s\), hence \(\mathrm {d}\bar{p} / \mathrm {d}\bar{\epsilon } =\left(s\,\mathrm {d}T\right)/\left(T\,\mathrm {d}s\right)\).

  21. 21.

    Note that this implies that the GPPZ flow [69] with superpotential \(W=-\frac{3}{4}\left(1+\cosh (\phi /\sqrt{3})\right)\) does not admit stable black-brane solutions below a certain minimum temperature because its potential asymptotes to \(V\rightarrow -\left(3/8\right)\exp (2\phi /\sqrt{3})\) for large \(\phi \).

  22. 22.

    Recall that \(V_{(1)}\left[\Delta _\mathrm {IR}\rightarrow 4+\right] = V_{(2)}\left[\gamma \rightarrow 0+\right]\), see Eqs. (4.108) and (4.112).

  23. 23.

    The same relations were found in Ref. [92] by coupling the fluid to external sources.

  24. 24.

    Consistently, \(H=0\) does not follow from the generalised Onsager relations, which were derived from an effective action for hydrodynamics in Refs. [96, 97] and which should apply to any uncharged conformal fluid.

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    Jonas Probst

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Probst, J. (2018). Second-Order Hydrodynamics and Universality in Non-conformal Holographic Fluids. In: Applications of the Gauge/Gravity Duality. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-93967-4_4

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